A unified matrix model including both CCA and F matrices in multivariate analysis: The largest eigenvalue and its applications

Let Z_M₁× N=T1/2X where (T1/2)²=T is a positive definite matrix and X consists of independent random variables with mean zero and variance one. This paper proposes a unified matrix model \[Ω=(ZU₂U₂TZT)⁻¹ZU₁U₁TZT,\] where U₁ and U₂ are isometric with dimensions N× N₁ and N×(N-N₂) respectively such that U₁TU₁=I_N₁, U₂TU₂=I_N-N₂ and U₁TU₂=0. Moreover, U₁ and U₂ (random or non-random) are independent of Z_M₁× N and with probability tending to one, rank(U₁)=N₁ and rank(U₂)=N-N₂. We establish the asymptotic Tracy–Widom distribution for its largest eigenvalue under moment assumptions on X when N₁,N₂ and M₁ are comparable. The asymptotic distributions of the maximum eigenvalues of the matrices used in Canonical Correlation Analysis (CCA) and of F matrices (including centered and non-centered versions) can be both obtained from that of Ω by selecting appropriate matrices U₁ and U₂. Moreover, via appropriate matrices U₁ and U₂, this matrix Ω can be applied to some multivariate testing problems that cannot be done by both types of matrices. To see this, we explore two more applications. One is in the MANOVA approach for testing the equivalence of several high-dimensional mean vectors, where U₁ and U₂ are chosen to be two nonrandom matrices. The other one is in the multivariate linear model for testing the unknown parameter matrix, where U₁ and U₂ are random. For each application, theoretical results are developed and various numerical studies are conducted to investigate the empirical performance.